In last year's Annual Report we described our search for a function of the shape of motoneuronal dendrites which increases for shapes more closely resembling natural dendrites. This would suggest a "goal" for the dendrite. To test this, model dendrites are compared to natural ones. The models are free to vary in branch lengths, diameters and angles, and branching pattern, but are constrained to natural rates of taper and the electrical properties of the membrane. We concluded that the shape function F should depend on the extracellular volume accessed. Local volumes within a distance R of the dendrite were weighted by the electrotonic coupling to the soma of the dendrites nearpoint, and summed to give weighted extracellular volume V. This was divided by its "cost," dendrite volume v, to give the shape function F =V/v. F was greater for branching structures, greater for dendrites which branch more near the soma, a property of motoneurons, and greater when the structures had approximately natural extent. This year we developed an efficient algorithm to search for optimum dendrites having largest F by varying the lengths, diameters and angles of all their branches. The lengths of branches of optimized model dendrites at successive branchpoints were similar to the natural lengths. The low order branches are shorter perhaps because they are thicker and cost more in volume. A plot of F against v for natural dendrites revealed that for small v they do not access the maximum possible V, but for the higher 2/3 of the natural range of v, they accessed as much or more tissue than our best model dendrites to date. This suggested that F is important for most dendrites, and that dendrites of different volumes v should be considered separately. For such an optimum function there should only be tendency toward optimum structures; lesser ones should also occur with a frequency that decreases with F. Thus the distributions of values of features of natural dendrites should resemble plots of F versus that value in optimized dendrites. For natural dendrites, a measure b of daughter to parent branch diameter at branch points is distributed around the value 1. For model dendrites F fell off for small b because v was too large. The peak value for dendrites varied with v but ranged around 1 in a distribution comparable to the natural one. We can now compare the distributions of F at various v for asymmetry of daughter diameters, sparsity of branching, stem diameter, and other features, to the corresponding natural frequency distributions. With Dr. T. G. Smith we have adapted measurements of a distribution of fractal dimensions ("multifractals") as a mathematical tool to express salient morphological features of cultured neurons and glia.